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- One-loop correction for $\phi \to e^+e^-$ under a Yukawa interaction seems to vanish trivially.
I am trying to calculate the amplitude for a decay under a Yukawa interaction to one-loop order (with massless fermions for simplicity).
If I'm not wrong, there are 4 diagrams that contribute to 1 loop, three diagrams involving self-energy corrections (i.e. inserting a loop into the external lines) and an extra diagram with vertex correction (a field exchanged by and ).
I have no problem calculating the integrals and using counterterms to cancel the infinities that arise, but I'm not sure if the conditions I use for renormalization are correct. Following the example of QED, to apply on-shell renormalization I used the following conditions;
The scalar propagator in the limit should be
The fermion propagator in the limit should be
The vertex function in the limit should be . ( is the momentum of the scalar particle.)
Now, because the self-energy diagrams are all in external legs, the first two corrections mean that those diagrams vanish.
But the third condition tells that the vertex correction must also vanish when the scalar particle is on-shell (as in my diagram). Therefore all the diagrams here vanish trivially due to renormalization conditions.
Is this analysis correct? Or did I make some mistake in the renormalization part?
If I'm not wrong, there are 4 diagrams that contribute to 1 loop, three diagrams involving self-energy corrections (i.e. inserting a loop into the external lines) and an extra diagram with vertex correction (a
I have no problem calculating the integrals and using counterterms to cancel the infinities that arise, but I'm not sure if the conditions I use for renormalization are correct. Following the example of QED, to apply on-shell renormalization I used the following conditions;
The scalar propagator in the limit
The fermion propagator in the limit
The vertex function in the limit
Now, because the self-energy diagrams are all in external legs, the first two corrections mean that those diagrams vanish.
But the third condition tells that the vertex correction must also vanish when the scalar particle is on-shell (as in my diagram). Therefore all the diagrams here vanish trivially due to renormalization conditions.
Is this analysis correct? Or did I make some mistake in the renormalization part?
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